What is (\partial)/(\partial x)(xe^{4x+2y}) ?

The answer to (\partial)/(\partial x)(xe^{4x+2y}) is e^{4x+2y}+4e^{4x+2y}x

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(\partial)/(\partial x)(xe^{4x+2y})

\frac{\partial}{\partial x} (x e^{4 x + 2 y})

e^{4 x + 2 y} + 4 e^{4 x + 2 y} x

Solution steps

\frac{\partial}{\partial x} (x e^{4 x + 2 y})

Treat

y

as a constant

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = e^{4 x + 2 y}

= \frac{\partial}{\partial x} (x) e^{4 x + 2 y} + \frac{\partial}{\partial x} (e^{4 x + 2 y}) x

\frac{\partial}{\partial x} (x) = 1

\frac{\partial}{\partial x} (e^{4 x + 2 y}) = e^{4 x + 2 y} \cdot 4

= 1 \cdot e^{4 x + 2 y} + e^{4 x + 2 y} \cdot 4 x

Simplify

= e^{4 x + 2 y} + 4 e^{4 x + 2 y} x

What is (\partial)/(\partial x)(xe^{4x+2y}) ?
The answer to (\partial)/(\partial x)(xe^{4x+2y}) is e^{4x+2y}+4e^{4x+2y}x